examples.sagews

#Example3.1

E=EllipticCurve([GF(3329)(0),0,0,49,0]); E;
E.is_ordinary();
E.cardinality();
t=E.frobenius().minpoly(); t
N=t.discriminant(); N
E.j_invariant();
EllipticCurve([GF(3329)(0),0,0,1,57]).j_invariant()
R=E.frobenius_order();
R;
R.degree();

Elliptic Curve defined by y^2 = x^3 + 49*x over Finite Field of size 3329 True 3280 x^2 - 50*x + 3329 -10816 1728 2492 Order in Number Field in phi with defining polynomial x^2 - 50*x + 3329 2

E=EllipticCurve([GF(3329^3)(0),0,0,1,378]); E;
E.is_ordinary();
E.cardinality();
t=E.frobenius().minpoly(); t
E.j_invariant();

Elliptic Curve defined by y^2 = x^3 + x + 378 over Finite Field in z3 of size 3329^3 True 36893154640 x^2 + 374350*x + 36892780289 189

####
E=EllipticCurve([GF(3329^3)(0),0,0,3,1152]); E;
E.is_ordinary();
E.cardinality();
t=E.frobenius().minpoly(); t
E.j_invariant();

Elliptic Curve defined by y^2 = x^3 + 3*x + 1152 over Finite Field in z3 of size 3329^3 True 36893154640 x^2 + 374350*x + 36892780289 1202

#Example 3.2
E0=EllipticCurve([GF(3329)(0),0,0,99,0]); E0;
E0.is_ordinary();
E0.cardinality();
t=E0.frobenius().minpoly(); t
N0=t.discriminant(); N0
#####
E1=EllipticCurve([GF(3329)(0),0,0,1,72]); E1;
E1.is_ordinary();
E1.cardinality();
t=E1.frobenius().minpoly(); t
N1=t.discriminant(); N1

Elliptic Curve defined by y^2 = x^3 + x + 72 over Finite Field of size 3329 True 3226 x^2 - 104*x + 3329 -2500

E3=EllipticCurve([GF(3329^3)(0),0,0,1,378]);
E4=EllipticCurve([GF(3329^3)(0),0,0,3,1152]);
E3.j_invariant()
E4.j_invariant()

189 1202

E0=EllipticCurve([GF(3329^3)(0),0,0,99,0]); E0;
E0.is_ordinary();
E0.cardinality();
t=E0.frobenius().minpoly(); t
N0=t.discriminant(); N0


E1=EllipticCurve([GF(3329^3)(0),0,0,1,192]); E1;
E1.is_ordinary();
E1.cardinality();
t=E1.frobenius().minpoly(); t
N1=t.discriminant(); N1

Elliptic Curve defined by y^2 = x^3 + 99*x over Finite Field in z3 of size 3329^3 True 36892694074 x^2 - 86216*x + 36892780289 -140137922500 Elliptic Curve defined by y^2 = x^3 + x + 192 over Finite Field in z3 of size 3329^3 True 36892694074 x^2 - 86216*x + 36892780289 -140137922500

E0=EllipticCurve([GF(3329)(0),0,0,99,0]); E0;
E0.is_ordinary();
E0.cardinality();
t=E0.frobenius().minpoly(); t
N0=t.discriminant(); N0

E=EllipticCurve([GF(3329^3)(0),0,0,49,0]); E;
E.is_ordinary();
E.cardinality();
t=E.frobenius().minpoly(); t

Elliptic Curve defined by y^2 = x^3 + 49*x over Finite Field in z3 of size 3329^3 True 36893154640 x^2 + 374350*x + 36892780289

################3

E=EllipticCurve([GF(3329^3)(0),0,0,1,98]); E;
E.is_ordinary();
E.cardinality();
t=E.frobenius().minpoly(); t

E.j_invariant()


elliptic_j(13*i)

Elliptic Curve defined by y^2 = x^3 + x + 98 over Finite Field in z3 of size 3329^3 True 36893154640 x^2 + 374350*x + 36892780289 3100 2.97704363274819e35